Let P and Q be distinct points on the parabola y2=8x such that a circle drawn on PQ as diameter touches the line x+2=0. If length of chord PQ is 2009 and the area of triangle PQR, (R is point of intersection of normals at P and Q on the parabola) is k sq.unit, then the value of 27k is

PQ is focal chord ∴t1t2=−1 a(t1+1t1)=2009⇒|t1+1t1|=103Area of ΔPQR=a22|t1+1t1|3=2×(103)3=200027

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Let P and Q be distinct points on the parabola $y^{2} = 8 x$ such that a circle drawn on PQ as diameter touches the line x+2=0. If length of chord PQ is $\frac{200}{9}$ and the area of triangle PQR, (R is point of intersection of normals at P and Q on the parabola) is k sq.unit, then the value of 27k is

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Detailed Solution

PQ is focal chord $∴ t_{1} t_{2} = - 1$
Question Image
$a (t_{1} + \frac{1}{t_{1}}) = \frac{200}{9} \Rightarrow | t_{1} + \frac{1}{t_{1}} | = \frac{10}{3}$
Area of $Δ P Q R = \frac{a^{2}}{2} {| t_{1} + \frac{1}{t_{1}} |}^{3} = 2 \times {(\frac{10}{3})}^{3} = \frac{2000}{27}$